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Statistics and Its Interface Volume 5 (2012) 345–354
Empirical likelihood inference for two-sample
problems
Changbao Wu∗ and Ying Yan
There exists a rich body of literature on empirical likelihood methods for two-sample problems. In this paper we
focus on the simple and yet very important case of making inference on the diﬀerence of two population means using the empirical likelihood approach. Our contributions to
this dynamic research topic include: (i) a weighted empirical likelihood method which not only performs well but also
has a major advantage in computational simplicity; (ii) a
pseudo empirical likelihood method for comparing two population means when the two samples are selected by complex surveys; (iii) two-sample empirical likelihood method
with missing responses; (iv) bootstrap calibration procedures for the proposed weighted and pseudo empirical likelihood methods. Results from a limited simulation study
showed that our proposed methods perform very well. The
methods are also applied to a real data example on family
expenditures.
AMS 2000 subject classifications: Primary 62G10;
secondary 62D05.
Keywords and phrases: Behrens-Fisher problem, Bootstrap calibration, Case-control studies, Conﬁdence intervals, Complex surveys, Hypothesis test, Nonparametric likelihood.
1. INTRODUCTION
Two-sample problems are commonly encountered in
many areas of statistics. In classical designed experiments
involving two levels of a single factor, the focus is on comparing two treatments under the controlled experimental settings. In case-control studies, two independent samples are
retrospectively taken, one from the case (or disease) group
and one from the control group, and the main interest is
to study relationships between a disease and environmental
or genetic characteristics (Qin, 1998). In observational studies, comparisons are often made between two groups deﬁned
by gender, age, ethnic backgrounds, educational levels, etc.
While the settings can be extremely simple, some of the
problems can be very interesting and fascinating. For example, Zhou, Gao and Hui (1997) studied the eﬀects of two
races on medical costs of patients. Their interest is whether
∗ Corresponding
author.
the average medical costs for African American patients is
the same as that for white patients. In the August 22, 2011
issue of Time magazine, Andrea Sachs reported diﬀerences
in earnings between the typical good-looking worker and
the below-average-looking worker over a lifetime, and his
reported results are indeed very surprising.
Suppose that Y11 , . . . , Y1n1 and Y21 , . . . , Y2n2 are two independent and identically distributed samples from Y1 and
Y2 , respectively, with E(Y1 ) = μ1 , E(Y2 ) = μ2 , V ar(Y1 ) =
σ12 and V ar(Y2 ) = σ22 . The well-known Behrens-Fisher problem is to test H0 : μ1 = μ2 against H1 : μ1 = μ2 when
both Y1 and Y2 are normally distributed with possibly unequal variances (Behrens, 1929; Fisher, 1935, 1939) or unknown ratio σ12 /σ22 (Ghosh and Kim, 2001). The classic
two-sample T-test under the normality assumption requires
that σ12 = σ22 . When both sample sizes n1 and n2 are
large and under the null hypothesis H0 , the test statistic
T = (Y¯2 − Y¯1 )/{S12 /n1 + S22 /n2 }1/2 follows approximately
n i
a standard normal distribution, where Y¯i = n−1
i
j=1 Yij
ni
2
−1
2
¯
and Si = (ni − 1)
j=1 (Yij − Yi ) , i = 1, 2. However,
when the sample sizes are small, normal approximation to
T becomes very poor, and a known better solution is to
use a t-distribution with the degree of freedom calculated
by the Welch-Satterthwaite equation (Welch, 1938, 1947;
Satterthwaite, 1946). When the distribution of Yij is not
normal but has a known form, speciﬁc adjustment to T can
be made to obtain more powerful tests. For instance, Zhou,
Gao and Hui (1997) studied cases where both Y1 and Y2
follow log-normal distributions, and they derived new tests
which showed substantial improvement over the T-test for
log-normally distributed sample data.
The empirical likelihood (EL) method was ﬁrst proposed
by Owen (1988) as a nonparametric likelihood-based alternative approach to inference on the mean of a single population. The approach has attracted immediate attention
from researchers since Owen’s original work, and applications of EL have been found in many areas of statistics. One
of the most signiﬁcant contributions to the EL methodology
was the work by Qin and Lawless (1994). They showed that
side information in the form of general estimating equations
can be eﬀectively incorporated into inference through constrained maximization of the empirical likelihood function.
Owen’s 2001 monograph provided an excellent overview of
the early developments on empirical likelihood. In recent
years, there have been several new areas where the empirical likelihood method has proved to be very useful. One of
these areas is on missing data problems; see, for instance,
Wang and Rao (2001, 2002a, 2002b), Wang and Veraverbeke
(2002), Wang, Linton and Hardle (2004), Qin and Zhang
(2007), Wang and Dai (2008), Zhou, Wan and Wang (2008),
Qin, Zhang and Leung (2009) and Wang and Chen (2009),
among others. Another area with substantial new development is the use of empirical likelihood for the analysis
of complex survey data; see, for example, Chen and Sitter
(1999), Wu (2004, 2005), Wu and Rao (2006, 2010), Kim
(2009) and Rao and Wu (2010a, 2010b), among others.
There has also been revived research on two-sample problems using the empirical likelihood method. Qin (1994) used
a semi-empirical likelihood approach to inference on the difference of two population means. Jing (1995) showed that
the two-sample empirical likelihood for the diﬀerence of two
population means is Bartlett correctable. Qin and Zhang
(1997), Qin (1998) and Zhang (2000) examined the EL
methods for the two-sample problems in the context of casecontrol studies. Claekens, Jing, Peng and Zhou (2003) derived empirical likelihood conﬁdence regions for the comparison distribution of two populations. They also considered
ROC curves which are used to compare measurements of a
diagnostic test from two populations. Zhou and Liang (2005)
studied empirical likelihood-based semiparametric inference
for the treatment eﬀect in the two-sample problem with
censoring. Cao and Van Keilegom (2006) developed an ELbased test on whether two populations follow the same distribution.
In this paper we focus on the simple and yet very important case of making inference on the diﬀerence of two
population means using the empirical likelihood approach.
Our contributions to this dynamic research topic are as follows: In Section 2, we propose a weighted empirical likelihood method which has a major advantage in computational
simplicity. In Section 3, we propose a pseudo empirical likelihood method for comparing means of two ﬁnite populations
when the two samples are selected by complex surveys. Twosample empirical likelihood with missing responses is discussed in Section 4. Bootstrap calibration procedures for the
weighted and pseudo empirical likelihood methods are presented in Section 5. Results from a limited simulation study
are reported in Section 6. A real data example is presented
in Section 7. Some additional remarks are given in Section 8.
2.1 Standard two-sample empirical likelihood
Let Y11 , . . . , Y1n1 and Y21 , . . . , Y2n2 be two independent
and identically distributed samples from Y1 and Y2 , respectively, with E(Y1 ) = μ1 , E(Y2 ) = μ2 , V ar(Y1 ) = σ12 and
V ar(Y2 ) = σ22 . Let θ = μ1 − μ2 be the parameter of interest.
The joint empirical log-likelihood function based on the two
samples is given by
(1)
l(p1 , p2 ) =
n1
log(p1j ) +
j=1
n2
log(p2j ),
j=1
where p1 = (p11 , . . . , p1n1 ) and p2 = (p21 , . . . , p2n2 ) are the
two sets of probability measure imposed respectively over
the two samples. The empirical log-likelihood ratio statistic
on the parameter of interest, θ, is deﬁned as
(2)
r(θ) =
n1
log{n1 pˆ1j (θ)} +
j=1
n2
log{n2 pˆ2j (θ)},
j=1
where pˆ1j (θ) and pˆ2j (θ) maximize l(p1 , p2 ) subject to the
following set of constraints:
n1
(3)
j=1
n1
(4)
n2
p1j = 1,
p2j = 1,
j=1
p1j Y1j −
j=1
n2
p2j Y2j = θ.
j=1
Let n = n1 + n2 be the combined sample size and θ =
E(Y1 ) − E(Y2 ).
Theorem 2.1. Suppose that σ12 < ∞, σ22 < ∞ and n1 /n →
π ∈ (0, 1) as n → ∞. Then −2r(θ) converges in distribution to a χ2 random variable with one degree of freedom as
n → ∞.
Proof. If n1 /n → π ∈ (0, 1) as n → ∞, then there is no need
−1/2
−1/2
to distinguish, for instance, among Op (n1 ), Op (n2 )
−1/2
and Op (n
). Let μ0 be a ﬁxed number such that μ0 =
μ2 + O(n−1/2 ). We replace constraint (4) by
(5)
n1
j=1
p1j Y1j = μ0 + θ
and
n2
p2j Y2j = μ0 .
j=1
The constraint (5) implies (4), but (4) does not imply (5).
The newly introduced μ0 is a nuisance parameter and will
eventually be proﬁled. It serves as a bridge for computing the EL ratio statistic for the parameter of interest, θ.
We start with the setting for the standard two-sample emLet r(μ0 , θ) be the empirical log-likelihood ratio statistic
pirical likelihood and provide a detailed proof of the asympwhen (4) is replaced by (5). The initial r(θ) corresponds
2
totic χ distribution of the empirical likelihood ratio statisto r(ˆ
μ0 , θ) where μ
ˆ0 is the maximum point of r(μ0 , θ) with
tic for the diﬀerence of two population means, which hasn’t
respect to μ0 for the ﬁxed θ.
been given in detail in any of the existing papers on emIt can be shown that p˜1i and p˜2j which maximize l(p1 , p2 )
pirical likelihood. More importantly, the proof shows the
subject to (3) and (5) are given by
computational diﬃculty involved in calculating the EL ratio statistic, which motivates the weighted EL approach we
1
(6)
p˜1j =
propose in section 2.2.
n1 {1 + λ1 (Y1j − μ0 − θ)}
2. WEIGHTED TWO-SAMPLE EMPIRICAL
LIKELIHOOD METHOD
346 C. Wu and Y. Yan
and
A (1−α)-level conﬁdence interval on θ can be constructed
p˜2j
1
,
=
n2 {1 + λ2 (Y2j − μ0 )}
where λ1 and λ2 are the solutions to
as
(12)
C1 = θ | −2r(θ) ≤ χ21 (α) ,
where χ21 (α) is the upper (100α)% quantile from the χ21 dis1
Y1j − μ0 − θ
tribution. The lower and upper bounds of the interval C1 re(7)
=0
quires
a bi-section search algorithm. This is a computationn1 j=1 1 + λ1 (Y1j − μ0 − θ)
ally challenging task, because for every selected grid point
on θ, one needs to maximize the EL ratio function over the
and
nuisance parameter, μ0 , and there is no-closed form solution
n
2
1 Y2j − μ0
to
the maximum point μ
ˆ0 for any given θ.
= 0.
n2 j=1 1 + λ2 (Y2j − μ0 )
The computational diﬃculties under the standard twosample EL formulation are due to the fact that the involved
The corresponding empirical likelihood ratio statistic is
Lagrange multipliers, which are determined through the set
given by
of equations (7), have to be computed based on two separate
n1
samples with an added nuisance parameter μ0 . Such diﬃ
r(μ0 , θ) = −
(8)
log{1 + λ1 (Y1i − μ0 − θ)}
culties can be avoided through an alternative formulation
i=1
of the EL function, for which computation procedures are
n2
virtually identical to those for one-sample EL problems.
−
log{1 + λ2 (Y2j − μ0 )}.
n1
j=1
To ﬁnd the maximum point of r(μ0 , θ) with respect to μ0 ,
we set ∂r(μ0 , θ)/∂μ0 = 0 which gives
2.2 Weighted two-sample empirical
likelihood
There exist diﬀerent versions of weighting schemes for the
empirical likelihood function. For instance, Ren (2008) dis(9)
n1 λ1 + n2 λ2 = 0.
cussed a weighted empirical likelihood approach for some
two-sample semiparametric models with various types of
Using standard argument as in Owen (2001, pages 219–222), censored data, where the weights depend on the censoring
we also have
type of individual observations. The method we use here is
to put a weight on each sample, which is related to the sam1 ¯
(10)
λ1 = 2 (Y1 − μ0 − θ) + op (n−1/2 )
ple size. This idea was ﬁrst used by Fu, Wang and Wu (2008)
σ1
when they discussed inferences with multiple samples.
and
We deﬁne the weighted empirical log-likelihood function
1 ¯
as
λ2 = 2 (Y2 − μ0 ) + op (n−1/2 ),
σ2
n1
n2
w1 w2 (13)
l
(p
,
p
)
=
log(p
)
+
log(p2j ),
w
1j
1
2
and this leads to the solution to (9) as
n1 j=1
n2 j=1
(11)
μ
ˆ0 = γ(Y¯1 − θ) + (1 − γ)Y¯2 + op (n−1/2 ),
where w1 = w2 = 1/2. This choice of w1 and w2 is to facilitate the reformulation of normalization constraints (3) and
where
the parameter constraint (4) into the alternative forms (14)
n1
n1
n2
γ=
+
/
.
and (15), to be speciﬁed below. Let the weighted EL ratio
2
2
2
σ1
σ1
σ2
statistic rw (θ) be deﬁned in the same way as r(θ) of Secˆ0 −θ) = op (1) and λ2 (Y2j −μ
ˆ0 ) = op (1) tion 2.1 under the same constraints (3) and (4) but replacNoting that λ1 (Y1i −μ
uniformly over all i and j and applying a Taylor series ex- ing l(p1 , p2 ) by the weighted version lw (p1 , p2 ). We have
pansion to (8) at μ0 = μ
ˆ0 , we have the following asymptotic the following result concerning the asymptotic distribution
expansion to the EL ratio function:
of rw (θ).
−2r(θ) = −2r(ˆ
μ0 , θ)
2 n2
n1 ¯
ˆ0 − θ + 2
= 2 Y1 − μ
σ1
σ
2 2
2
σ
1
= Y¯1 − Y¯2 − θ /
+
n1
2
Y¯2 − μ
ˆ0 + op (1)
σ22
+ op (1).
n2
It immediately follows that −2r(θ) converges in distribution
to a χ2 random variable with one degree of freedom.
Theorem 2.2. Suppose that σ12 < ∞, σ22 < ∞ and n1 /n →
π ∈ (0, 1) as n → ∞. Then −2rw (θ)/c1 converges in distribution to a χ2 random variable with one degree of freedom
as n → ∞, where c1 is a scaling constant and is speciﬁed
in (17).
Proof. The most crucial step we use to prove the result
is a reformulation of the constraints, which also characterizes the required computational procedures for our proposed
Empirical likelihood inference for two-sample problems 347
weighted EL. First, we note that
two normalization
the
2 conn1
straints in (3) are equivalent to j=1
p1j = 1 and i=1 wi ×
n i
(4) induced by the
j=1 pij = 1. Second, the constraint n1
parameter θ can be re-written as w1 j=1
p1j (Y1j /w1 ) +
n 2
w2 j=1 (−Y2j /w2 ) = θ. The two sets of constraints (3)
and (4) can now be equivalently written as
(14)
2
wi
i=1
(15)
2
ni
pij = 1,
j=1
wi
ni
pij uij = 0,
= U D −1 U + op n−1
2
= d(22) Y¯1 − Y¯2 − θ + op n−1 ,
where d(22) is the second diagonal element of D −1 . If we let
2
S1
S22
(22)
(17)
c1 = d
+
,
n1
n2
then it follows immediately that −2rw (θ)/c1 converges in
distribution to a χ2 random variable with one degree of freedom when θ = E(Y1 ) − E(Y2 ).
The weighted two-sample EL formulation is computationally friendly, since it does not involve any nuisance paramewhere uij = Z ij − η, Z 1j = (1, Y1j /w1 ) , Z 2j = (0, −Y2j / ters, and the involved equation (16) for the Lagrange multiw2 ) , η = (w1 , θ) . It should be noted that the choice w1 = plier can be solved using the one-sample EL algorithm (Wu,
w2 = 1/2 is important for the above reformulation of con- 2004). The scaling constant c1 involves the unknown θ and
straints. Using the standard Lagrange multiplier method, it can be consistently estimated by cˆ1 when θ is estimated by
can be shown that the pij which maximize lw (p1 , p2 ) sub- θˆ = Y¯1 − Y¯2 . The resulting −2rw (θ)/ˆ
c1 still has an asympject to (14) and (15) are given by pij = 1/{ni (1 + λ uij )}, totic χ21 distribution. Consequently, a (1−α)-level conﬁdence
and the Lagrange multiplier λ is the solution to
interval on θ can be constructed as
ni
2
c1 ≤ χ21 (α) .
(18)
C2 = θ | −2rw (θ)/ˆ
wi uij
(16)
g1 (λ) =
= 0.
n
1 + λ uij
i=1 i j=1
The scaling constant c1 can also be bypassed through a bootstrap
calibration method. See Section 5 for further detail.
Substituting 1/(1 + λ uij ) = 1 − λ uij /(1 + λ uij ) into (16),
we have
3. TWO-SAMPLE PSEUDO EMPIRICAL
⎛
⎞
ni
ni
2
2
LIKELIHOOD FOR COMPLEX SURVEY
uij uij
wi
wi
⎝
⎠λ =
uij .
DATA
n
n
1 + λ uij
i=1 i j=1
i=1 i j=1
i=1
j=1
In this section, we use a pseudo empirical likelihood
(PEL) approach to develop a method which is suitable for
two-sample problems involving complex survey data. The
ni
2
wi PEL method was ﬁrst proposed by Chen and Sitter (1999)
U=
uij = 0, Y¯1 − Y¯2 − θ ,
for point estimation, and was later further developed by Wu
n
i=1 i j=1
and Rao (2006) for conducting hypothesis tests or constructand using a similar argument as in Owen (2001, pages 219 − ing conﬁdence intervals based on a single complex survey
sample.
222), we get
Let si be the set of sampled units selected from the ith
λ = D −1 U + op n−1/2 ,
ﬁnite population, i = 1, 2. Let n1 and n2 be the respective
(i)
sample size. Let πj = P (j ∈ si ) be the inclusion probabiliwhere
Noting that
D=
ni
2
wi i=1
ni
j=1
(i)
uij uij
ties, j = 1, . . . , ni and i = 1, 2. Let dij = 1/πj be the basic
design weights and
dik
d˜ij (si ) = dij /
is a 2 × 2 matrix. The weighted two-sample empirical logk∈si
likelihood ratio statistics for θ can now be expanded as follows:
be the normalized design weights. We follow design-based
framework for inference, where the value Yij for the study
n
2
i
wi variable Y , attached to the jth unit in population i, is
−2rw (θ) = 2
log(1 + λ uij )
n
treated as ﬁxed, and randomization is induced by the rani=1 i j=1
dom selection of sampled units. Let μ1 and μ2 be the respec
ni 2
−1 wi 1 tive
ﬁnite population means. Let θ = μ1 − μ2 be the param=2
λ uij − λ uij uij λ + op n
n
2
eter of interest. The two-sample pseudo empirical likelihood
i=1 i j=1
348 C. Wu and Y. Yan
function is deﬁned as
(19)
lpel (p1 , p2 ) = w1
n1
d˜1j (s1 ) log(p1j )
j=1
+ w2
n2
d˜2j (s2 ) log(p2j ),
and assume that n1 /n → π ∈ (0, 1) as n → ∞, the adjusted two-sample pseudo empirical likelihood ratio statistic −2rpel (θ)/c2 converges in distribution to a χ2 random
variable with one degree of freedom as n → ∞, where
θ = μ1 − μ2 .
The deﬁnition of the scaling constant c2 involves the unknown θ and variances of the two H´ajek estimators of the
where w1 = w2 = 1/2. When the selection probabilities population means. Let cˆ2 be a consistent estimator of c2 .
(i)
πj are all equal for the given population i, the normalized The (1 − α)-level PEL conﬁdence interval on θ can be constructed as
design weights d˜ij (si ) reduces to 1/ni . In this case our pro
posed pseudo empirical likelihood function lpel (p1 , p2 ) rec2 ≤ χ21 (α) .
(21)
C3 = θ | −2rpel (θ)/ˆ
duces to the weighted EL function lw (p1 , p2 ) that has been
discussed in Section 2.2.
The maximumPEL estimator of
In Section 5, we propose a two-sample bootstrap calibration
n1
n2
pˆ1j Y1j − j=1
pˆ2j Y2j , where
θ is computed as θˆpel = j=1
procedure which can bypass the need for calculating the
the pˆij ’s maximize lpel (p1 , p2 ) subject to the normalization
scaling constant under certain sampling designs.
constraints (3), and the estimator is given by θˆpel = μ
ˆ1 − μ
ˆ2 ,
n i ˜
n i
n i
where μ
ˆi = j=1 dij (si )Yij = j=1 dij Yij / j=1 dij is the
H´
ajek estimator of the ﬁnite population mean, μi . Under 4. TWO-SAMPLE EMPIRICAL LIKELIHOOD
WITH MISSING DATA
complex survey designs, the estimator θˆpel is approximately
design-unbiased, while the regular estimator θˆ = Y¯1 − Y¯2 is
In this section, we extend the discussion to two-sample
biased.
problems where measures on certain auxiliary variables are
Let pˆij (θ) be the maximizer of lpel (p1 , p2 ) subject to the available for all units in both samples but values of the renormalization constraints (3) and the constraint (4) induced sponse variable are subject to missingness. Suppose that
by the parameter of interest, θ. It follows from similar argu{(Yij , xij ), j = 1, . . . , ni } is a conceptual random sample
ments in Section 2.2 that pˆij (θ) = d˜ij (si )/(1 + λ uij ), where
from the ith population, i = 1 and 2, where the dimensionthe uij is deﬁned in Section 2.2, both λ and uij depend on
ality and the components in the vector of auxiliary variables
θ, and the Lagrange multiplier λ is the solution to
xij can be diﬀerent for the two populations but measures on
n
2
xij are observed for all j. The response variable Yij could
i
d˜ij (si )uij
be missing. Let δij = 1 if Yij is observed, and δij = 0 if
wi
= 0.
(20)
g2 (λ) =
1 + λ uij
i=1
j=1
Yij is missing. The two actual samples may be represented
by {(Yij , xij , δij ), j = 1, . . . , ni }, i = 1, 2. We assume that
Noting that pˆij (θˆpel ) = d˜ij (si ), the two-sample pseudo em- P (δij = 1 | Yij , xij ) = P (δij = 1 | xij ). That is, the repirical log-likelihood ratio statistic for θ is given by
sponses are missing-at-random. We further assume that the
following linear regression models hold:
ˆ 2 (θ) − lpel p
ˆ 2 (θˆpel )
ˆ 1 (θ), p
ˆ 1 (θˆpel ), p
rpel (θ) = lpel p
(22)
Yij = xij β i + ij , j = 1, . . . , ni , i = 1, 2,
ni
2
=−
wi
d˜ij (si ) log(1 + λ uij ).
where β 1 and β 2 are the regression parameters for the two
i=1
j=1
populations, and the ij ’s are independent error terms with
n i ˜
2
mean
0 and unknown variance τi2 . The parameter of interest
Let K = i=1 wi j=1 dij (si )uij uij , which is a 2×2 matrix
is
still
θ = μ1 − μ2 , where μi = E(Yij ) is the unconditional
involving the unknown θ, and
mean response for the ith population.
⎧ n
⎫
n1
2
⎬
⎨
It should be noted that when complete responses are
,
d˜1j (s1 )Y1j + V
d˜2j (s2 )Y2j
c2 = k (22) V
available,
measures of the auxiliary variables xij provide
⎭
⎩
j=1
j=1
no additional information for the unconditional population
means μ1 and μ2 . When the Yij ’s are subject to missingness,
where k (22) is the second diagonal element of K −1 , and V (·)
information collected on xij becomes valuable and can be
refers to design-based variance. The following result can now
used to impute the missing responses through the regression
be established using arguments similar to the proof of Themodel (22). Let
orem 2.2. Details are omitted.
⎞−1
⎛
Theorem 3.1. Under the asymptotic framework and the
ni
ni
⎠
˜ =⎝
regularity conditions C1–C3 described in Wu and Rao
β
δ
x
x
δij xij Yij
ij ij ij
i
(2006), applicable to both populations and sampling designs,
j=1
j=1
j=1
Empirical likelihood inference for two-sample problems 349
be the ordinary least square estimator of β i using available distribution to a χ2 random variable with one degree of freesample data. Let
dom. The details can be found in Yan (2010) and are omitted
here.
Y˜ij = δij Yij + (1 − δij )xij β˜i , j = 1, . . . , ni , i = 1, 2.
In practice, the scaling constant c3 needs to be estimated
Note that Y˜ij = Yij is the observed response if δij = 1, and in order to use the above result for testing hypotheses or
˜ is the regression imputed value for Yij if δij = 0. constructing conﬁdence intervals on θ. This can be done by
Y˜ij = xij β
i
We consider the standard two-sample empirical likelihood using plug-in moment estimators for Sik , μi , k = 1, . . . , 6,
formulation described in Section 2.1. Let r˜(θ) be deﬁned in i = 1, 2, the least square estimators of β i , and the estimathe same way as r(θ) given by (2), with constraint (4) being tor of τi2 from the ﬁtted residuals. It is also possible to dereplaced by
rive a similar result under the weighted empirical likelihood
formulation of Section 2.2. or a result in the same spirit of
n1
n2
Theorem 3.1. when the two samples are selected by complex
˜
˜
p1i Y1i −
p2j Y2j = θ.
(23)
survey designs and the responses are subject to missingness.
i=1
j=1
These scenarios will not be discussed further in the current
We assume that (Yij , xij , δij ), j = 1, . . . , ni can be viewed paper.
as an independent and identically distributed sample from
(Yi , X i , δi ), i = 1, 2.
5. BOOTSTRAP PROCEDURES
Theorem 4.1. Suppose that the two regression models speciﬁed by (22) hold, and the error variances τi2 are ﬁnite. Assume that E(X i 2 ) < ∞, i = 1, 2, n1 /n → π ∈ (0, 1)
as n → ∞. Then −2˜
r(θ)/c3 converges in distribution to a
χ2 random variable with one degree of freedom as n → ∞,
where θ = μ1 − μ2 and c3 is a scaling constant and is speciﬁed in (24).
Empirical likelihood-based hypothesis tests or conﬁdence
intervals on θ rely on the asymptotic distribution of the
empirical likelihood ratio statistic. For single samples and
when the sample sizes are not large, bootstrap calibration
methods often provide improved performance under standard settings. For the weighted or the pseudo empirical likelihood methods for two-sample problems, there is an added
Proof. The imputation procedure used here is the same as value to the bootstrap methods: the scaling constants c1 or
the one used by Wang and Rao (2002b) for a single sample. c2 , which depends on unknown population quantities and
The major technical diﬃculty is that the Y˜ij ’s are not inde- needs to be estimated, can be bypassed. Simulation stud˜ for all imputed values. Under ies also show that the bootstrap calibrated empirical likependent due to the use of β
i
the assumed conditions and following the same arguments lihood methods provide improved performance for samples
used by Wang and Rao (2002b), we have
with small or moderate sizes.
We now present a bootstrap calibration procedure for the
ni
d
1 two-sample
pseudo empirical likelihood method described in
˜
Yij − μi → N (0, Vi )
√
ni j=1
Section 3. The proposed procedure also covers the weighted
empirical likelihood approach of Section 2.2 as a special
and
case with equal probability selection of units. Our pron
i
posed procedure is modiﬁed from the one-sample bootstrap
2 p
1 ˜
Yij − μi → Ui ,
method described in Wu and Rao (2010). The two-sample
ni j=1
pseudo empirical likelihood ratio conﬁdence interval (24)
on θ, which involves the scaling constant c2 and the upwhere
per α-quantile from a χ21 distribution, can be replaced by
−1
Vi = Si1 + Si2
Si3
Si2 τi2 + β i Si4 β i
{θ | rpel (θ) > bα }, where bα is the α-quantile of the sampling
−1
distribution of the pseudo empirical likelihood ratio statis− 2Si5
β i μi + μ2i + 2Si2
Si3
Si6 τi2 ,
tic, rpel (θ). The value of bα in (25) can be approximated
β i μi + μ2i ,
Ui = Si1 + β i Si4 β i − 2Si5
through the following bootstrap calibration procedure. Two
and Si1 = E{δi (Yi − X i β i )2 }, Si2 = E{(1 − δi )X i }, Si3 = important ingredients of the proposed bootstrap procedure
E(δi X i X i ), Si4 = E(X i X i ), Si5 = E(X i ), Si6 = E(δi X i ). are (i) the basic design weights dij need to be treated as part
of the sample data; and (ii) a bootstrap version of the twoIf we deﬁne
sample pseudo empirical likelihood function should be used.
V1
U1
V2
U2
(24)
c3 =
+
+
/
,
1. Select a bootstrap sample s∗i of size ni from the orign1
n2
n1
n2
inal sample si using simple random sampling with replacement and denote the bootstrap sample data by
then it follows from similar arguments used in the proof
of Theorem 2.1 of Section 2.1 that −2˜
r(θ)/c3 converges in
{(d∗ij , Yij∗ ), j ∈ s∗i }, i = 1, 2.
350 C. Wu and Y. Yan
2. Let the bootstrap version of the two-sample pseudo empirical likelihood function be deﬁned as
∗
(p1 , p2 ) = w1
d˜∗1j (s∗1 ) log(p1j )
lpel
interval C3∗ under the proposed with-replacement bootstrap
procedure tends to have an over-coverage problem, as shown
by the one-sample simulation results reported in Wu and
Rao (2010).
j∈s∗
1
+ w2
d˜∗2j (s∗2 ) log(p2j ),
6. SIMULATION STUDIES
j∈s∗
2
In this section we report results from a limited simulation study on the ﬁnite sample performance of the proposed
where d˜∗ij (s∗i ) = d∗ij / j∈s∗ d∗ij .
empirical likelihood ratio conﬁdence intervals on the difi
3. Calculate the bootstrap version of the two-sample ference of two population means, with comparison to the
pseudo empirical likelihood ratio statistic as
conventional T-test based method. We consider three cases:
(i) Both Y1 and Y2 are normally distributed; (ii) Both Y1
p1j
∗
and Y2 follow lognormal distributions; (iii) The two samples
rpel
(θˆpel ) = w1
d˜∗1j (s∗1 ) log ∗ ∗
d˜1j (s1 )
are selected from two ﬁnite populations, with the response
j∈s∗
1
variable containing many zero values. More scenarios of sim
p2j
ulations, not reported here to save space, can be found in
d˜∗2j (s∗2 ) log ∗ ∗ ,
+ w2
d˜2j (s2 )
Yan (2010).
j∈s∗
2
For cases (i) and (ii), two independent and identically dis∗
(p1 , p2 ) subject to
where the pij ’s maximize lpel
tributed samples of sizes n1 and n2 are drawn respectively
from Y1 and Y2 , and four conﬁdence intervals on θ = μ1 −μ2
n1
n2
are computed: (1) the conventional normal-approximation
p1j = 1,
p2j = 1,
interval based on T = {(Y¯1 − Y¯2 ) − (μ1 − μ2 )}/{S12 /n1 +
j=1
j=1
S22 /n2 }1/2 (T-Test); (2) the standard two-sample empirical
n1
n2
likelihood method described in Section 2.1 (EL); (3) the
∗
∗
p1j Y1j
−
p2j Y2j
= θˆpel ,
weighted empirical likelihood method introduced in Secj=1
j=1
tion 2.2 (WEL); and (4) the bootstrap-calibrated weighted
n i ˜
empirical likelihood method (BWEL). The nominal value of
ˆ
where θpel = μ
ˆ1 − μ
ˆ2 and μ
ˆi = j=1 dij (si )Yij .
4. Repeat Steps 1, 2 and 3 a large number of times, the conﬁdence level is ﬁxed at 95%. Performances of these
B, independently, to obtain the sequence r1∗ (θˆpel ), . . . , intervals are evaluated based on coverage probability (CP ),
lower (L) and upper (U ) tail error rates and average length
∗ ˆ
rB
(θpel ). Let b∗α be the 100αth sample quantile from
(AL), computed as follows:
this sequence.
The bootstrap calibrated two-sample pseudo empirical
likelihood ratio conﬁdence interval on θ can now be constructed as
(25)
C3∗ = {θ | rpel (θ) > b∗α } .
L = 100 ×
B
1 ˆ(b)
I θL ≥ θ ,
B
b=1
B
1 ˆ(b)
U = 100 ×
I θU ≤ θ ,
B
b=1
The following theorem states that this interval has the desired level of coverage probability. The proof is a combination of arguments used in the proof of Theorem 3.1 and
those used in the proof of Theorem 1 of Wu and Rao (2010). and
Details are omitted.
B
1 ˆ(b)
(b) CP = 100 ×
I θL < θ < θˆU
B
b=1
B
1 ˆ(b) ˆ(b) Theorem 5.1. The bootstrap calibrated two-sample pseudo
AL =
θU − θL ,
B
empirical likelihood ratio conﬁdence interval C3∗ on θ has
b=1
asymptotically correct coverage probability at (1 − α)-level if
(b) (b)
where (θˆL , θˆU ) is a conﬁdence interval on θ computed from
the conditions of Theorem 3.1 hold and the original samples
the bth simulated sample, and B = 2, 000 is the total nums1 and s2 are selected using unequal probability sampling
ber of simulation runs. For the bootstrap-calibrated method,
with replacement.
the number of bootstrap samples used for computing the inIn practice, complex survey samples are usually selected terval is 1, 000. It should be noted that L + CP + U = 100
by without-replacement sampling procedures. The interval for any method.
C3∗ can be used for without-replacement sampling designs if
Table 1 summarizes the results for case (i), where Y1 ∼
the sampling fraction fi = ni /Ni is negligible. Here Ni is the N (μ1 , σ12 ) and Y2 ∼ N (μ2 , σ22 ), μ1 = μ2 = 1, σ1 = 1.5 and
size of the ith ﬁnite population. When fi is not small, the σ2 = 1. We have the following major observations: (a) The
Empirical likelihood inference for two-sample problems 351
Table 1. Conﬁdence Intervals on θ = μ1 − μ2 for Two
Normal Populations
(n1 , n2 )
(30,30)
(30,60)
(60,30)
(60,60)
(30,90)
(90,30)
CI
T-Test
EL
WEL
BWEL
T-Test
EL
WEL
BWEL
T-Test
EL
WEL
BWEL
T-Test
EL
WEL
BWEL
T-Test
EL
WEL
BWEL
T-Test
EL
WEL
BWEL
L
2.10
2.45
2.20
2.05
4.70
3.10
2.75
2.70
0.80
2.55
2.20
2.40
2.40
2.50
2.45
2.55
4.65
2.95
2.75
2.60
1.00
2.35
2.30
2.35
CP
95.05
94.40
94.90
95.25
91.20
94.20
94.60
94.75
98.20
94.90
95.55
95.10
94.95
94.60
94.70
94.85
89.85
94.45
94.75
94.90
97.95
94.60
94.70
94.65
U
2.85
3.15
2.90
2.70
4.10
2.70
2.65
2.55
1.00
2.55
2.25
2.50
2.65
2.90
2.85
2.60
5.50
2.60
2.50
2.50
1.05
3.05
3.00
3.00
AL
1.31
1.28
1.30
1.31
1.06
1.18
1.20
1.21
1.21
1.04
1.05
1.05
0.92
0.91
0.92
0.92
0.95
1.14
1.16
1.16
1.16
0.94
0.95
0.95
T-test method provides excellent results for scenarios where
n1 = n2 . Coverage probabilities are very close to the nominal
value, and the two tail error rates are balanced; (b) When
the two sample sizes n1 and n2 are not equal, the T-test
method performs poorly, and the interval is either too wide
or too narrow, depending on which sample is bigger; (c) The
empirical likelihood-based method is not sensitive to the unequal sample sizes, and the weighted EL performs uniformly
better than the standard two-sample EL method. The coverage probabilities are good, the two tail error rates are very
balanced, and the average length is not inﬂated for all cases;
(d) The bootstrap-calibrated weighted EL method also provides excellent results.
Results for case (ii) where Y1 ∼ Lognormal(ν1 , σ12 ), Y2 ∼
Lognormal(ν2 , σ22 ), ν1 = 1.1, ν2 = 1.2, σ12 = 0.4 and σ22 =
0.2 are summarized in Table 2. All major points observed
from case (i) still hold, except that the performances of the
EL-based method seem to have deteriorated a little bit for
the case n1 = 30 and n2 = 90. The bootstrap-calibrated EL
method provides acceptable results for all cases.
For case (iii), we consider two ﬁnite populations of
sizes N1 = N2 = 5, 000. For the ﬁrst population, M1 =
3, 000 responses Y1j are set to zero, and for the second population, M2 = 4, 000 responses Y2j are zero. The
nonzero responses are generated from Uniform(0.8, 1.2) and
Uniform(1.8, 2.2), respectively. The two samples are taken
by simple random sampling without replacement. Under
such sampling designs, the pseudo-EL method of Section 3
reduces to the weighted EL method of Section 2.2. The re352 C. Wu and Y. Yan
Table 2. Conﬁdence Intervals on θ = μ1 − μ2 for Two
Lognormal Populations
(n1 , n2 )
(30,30)
(30,60)
(60,30)
(60,60)
(30,90)
(90,30)
CI
T-Test
EL
WEL
BWEL
T-Test
EL
WEL
BWEL
T-Test
EL
WEL
BWEL
T-Test
EL
WEL
BWEL
T-Test
EL
WEL
BWEL
T-Test
EL
WEL
BWEL
L
1.05
2.55
2.40
1.50
2.90
2.20
1.65
0.95
1.00
2.95
3.25
2.50
1.25
2.80
2.65
2.00
4.90
2.55
1.75
1.10
0.15
2.35
3.15
2.28
CP
93.10
91.85
92.25
93.75
89.45
93.20
93.15
94.40
97.35
93.45
94.50
95.65
93.15
92.60
93.00
94.05
86.10
92.20
91.85
93.05
97.75
93.75
94.60
95.00
U
5.85
5.60
5.35
4.75
7.65
4.60
5.20
4.65
2.55
3.60
2.25
1.85
5.60
4.60
4.35
3.95
9.00
5.25
6.40
5.85
2.10
3.90
2.25
2.20
AL
2.65
2.68
2.73
3.07
2.06
2.50
2.55
2.77
2.52
2.13
2.16
2.32
1.87
1.92
1.94
2.08
1.82
2.42
2.48
2.64
2.46
1.89
1.92
2.03
Table 3. Conﬁdence Intervals on θ = μ1 − μ2 for Two Finite
Populations
(n1 , n2 )
(30,30)
(30,60)
(60,30)
(60,60)
(30,90)
(90,30)
CI
T-Test
EL
WEL
BWEL
T-Test
EL
WEL
BWEL
T-Test
EL
WEL
BWEL
T-Test
EL
WEL
BWEL
T-Test
EL
WEL
BWEL
T-Test
EL
WEL
BWEL
L
4.40
3.10
2.70
2.30
2.25
2.95
2.20
2.15
6.70
3.40
4.30
3.10
3.20
2.15
2.05
2.00
1.00
2.15
1.60
1.60
7.65
2.85
4.00
2.60
CP
94.25
94.70
95.30
95.65
97.15
94.45
94.95
94.90
90.00
94.25
93.80
95.05
95.35
95.25
95.55
95.45
98.75
95.40
95.05
94.90
86.65
94.55
94.10
95.35
U
1.35
2.20
2.00
2.05
0.60
2.60
2.85
2.95
3.30
2.35
1.90
1.85
1.45
2.60
2.40
2.55
0.25
2.45
3.35
3.50
5.70
2.60
1.90
2.05
AL
0.68
0.66
0.67
0.68
0.63
0.53
0.54
0.54
0.54
0.61
0.62
0.63
0.48
0.47
0.48
0.48
0.62
0.48
0.49
0.48
0.49
0.59
0.60
0.61
sults are reported in Table 3. The T-test method once again
is very sensitive to the unequal sample sizes and performs
Table 4. Conﬁdence Intervals on the Diﬀerences in Family Expenditures
Response
Clothing (Y )
CI
T-Test
EL
WEL
BWEL
D12
(−0.194, 0.364)
(−0.197, 0.366)
(−0.203, 0.362)
(−0.207, 0.365)
D13
(1.138, 1.594)
(1.146, 1.605)
(1.134, 1.597)
(1.123, 1.607)
D23
(1.073, 1.490)
(1.081, 1.506)
(1.073, 1.498)
(1.081, 1.489)
Total (Z)
T-Test
EL
WEL
BWEL
(−7.680, 0.497)
(−7.630, 0.566)
(−7.661, 0.541)
(−7.700, 0.581)
(21.614, 28.623)
(21.696, 28.732)
(21.563, 28.637)
(21.437, 28.760)
(25.744, 31.676)
(25.757, 31.704)
(25.686, 31.657)
(25.682, 31.661)
poorly with unequal sample sizes. All three EL-based meth- ﬁnite populations using complex survey designs. One of the
ods, however, provide good and robust results for all scenar- major advantages of our proposed methods is the computational simplicity. Conﬁdence intervals can be constructed
ios considered in the simulation.
using algorithms developed for one-sample problems and no
nuisance parameters are involved. Under the scenarios ex7. A REAL DATA EXAMPLE
amined in the simulation study, our proposed methods proWe now apply the proposed weighted EL method to an- vide better results than the T-test method or the standard
alyzing the data from the 2000 Statistics Canada’s Family two-sample EL method. We are currently studying the perExpenditure Survey for the province of Ontario. The data formance of the pseudo EL method for unequal probability
set consists of N = 2, 248 observations on variables related multistage sampling designs, as well as methods for missing
to family compositions, income and expenditures, including data problems.
Y : the annual expenditure on clothing and Z: the total anThe empirical likelihood method is usually eﬀective and
nual expenditure. One of the research questions is to see powerful in dealing with populations with skewed distribuhow family expenditures are related to the number of chil- tions. For one sample problems, Chen, Chen and Rao (2003)
dren (age ≤ 15) in the family. The original survey used a and Chen and Qin (2003) applied the empirical likelihood
stratiﬁed simple random sampling design, but unfortunately method to populations with non-negative responses and a
the strata weight information is not available. In what fol- large portion of zero values. They showed that the empirical
lows, we ignore the stratiﬁcation and treat the data as if likelihood method is extremely eﬃcient. The same scenario
they were collected by simple random sampling.
also applies to two-sample problems. Hallstrom (2010) deWe break the data set into three subsets, corresponding veloped a modiﬁed Wilcoxon test for comparing two poputo families with two or more children, one child or no chil- lations with non-negative distributions and clumps of zeros,
dren. The subsample sizes for the three groups are n1 = 428, where parametric approaches are problematic. Our proposed
n2 = 579 and n3 = 1, 241, respectively. For a given response two-sample empirical likelihood methods can be directly apvariable (Y or Z), let μi , i = 1, 2, 3, be the unknown pop- plied to such scenarios and our limited simulation results
ulation means. We are interested in the pairwise diﬀerences show that the EL-based methods can be very promising.
Dii = μi − μi . For each diﬀerence, we compute 95% conﬁdence intervals using the four methods described in SecACKNOWLEDGEMENT
tion 6: T-Test, EL, WEL and BWEL. The results are given
This research was supported by grants to C. Wu from
in Table 4 (values in $1,000’s). It is a bit surprising to see
the Natural Sciences and Engineering Research Council of
that all four methods provide very similar results. This is
Canada and Mathematics of Information Technology and
partially due to the relatively large sample sizes for all the
Complex Systems. The authors thank the Editor, an Asthree groups. Conﬁdence intervals for D12 contain 0, which
sociate Editor and a reviewer for many helpful comments
implies that there is no signiﬁcant diﬀerence in spending bewhich led to improved presentation of the paper.
tween families with one child and families with two or more
children. Families with one or more children spend at least
Received 3 October 2011
$1,000 more on clothing and at least $21,000 more in total
expenditure than families with no children.
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Changbao Wu
Department of Statistics and Actuarial Science
University of Waterloo
Waterloo, ON N2L 3G1
Canada
E-mail address: [email protected]
Ying Yan
Department of Statistics and Actuarial Science
University of Waterloo
Waterloo, ON N2L 3G1
Canada
E-mail address: [email protected]